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Homeland Security and missile defense technology make it paramount that we be able to detect incoming projectiles or missiles. To make the defense successful, multiple radar screens are: required. Suppose it is determined that three independent screens are to be operated and the probability that any one screen will detect an incoming missile is 0.8. Obviously, if no screens detect an incoming projectile, the system is unworthy and must be improved. (a) What is the probability that an incoming missile will not be detected by any of the three screens? (b) What is the probability that the missile will be detected by only one screen? (e:) What is the probability that it. will be detected by at least two out of three screens?

Step by step solution

01

Establish the Binomial Distribution Parameters

Let 'p' represent the probability of detection by any one screen, which is \(p = 0.8\). Let 'n' represent the total number of screens, which is \(n = 3\).

02

Calculate the Probability of No Detection

The probability of a missile not being detected by any of the screens will be \( (1 - p)^n = (1 - 0.8)^3 = 0.008 \). The probability is the product because the probabilities of detection on each screen are independent of each other.

03

Calculate the Probability of Detection by Only One Screen

The probability of a missile being detected by only one screen can be calculated by a binomial distribution formula, which is \(\binom{n}{k} * (p)^k * (1 - p)^{n-k}\). Here, 'k' is the number of successful detections, i.e., 1. Substituting the values, we get \(\binom{3}{1} * (0.8)^1 * (1 - 0.8)^{3-1} = 0.096\).

04

Calculate the Probability of Detection by at Least Two Screens

The probability of a missile being detected by at least two screens would be 1 minus the sum of probabilities of no detection and detection only by one screen. Calculating this gives us \(1 - ((1 - p)^n + \binom{n}{1} * p * (1 - p)^{n-1}) = 1 - (0.008 + 0.096) = 0.896\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is a branch of mathematics focused on analyzing random phenomena. It assesses how likely events are to happen. In our scenario, we consider the use of probability to predict the performance of radar screens detecting incoming missiles. The main idea is to determine the likelihood of different outcomes based on known probabilities, like each screen having a detection probability of 0.8. This probability value suggests that each radar screen should, on average, detect 8 out of 10 incoming objects. In probability theory, we often work with situations involving multiple trials. For example, with three radar screens, we examine what the cumulative probability is for a given number of detections.

Independent Events

In probability, events are considered independent if the outcome of one event does not affect the outcome of another. For our exercise, each radar screen's attempt to detect a missile is independent.

• The chance of detection by one radar does not influence another.
• This independence allows us to calculate the combined probabilities using multiplication principles.

For instance, if each radar has an independent detection probability of 0.8, then the probability of all failing to detect an object is calculated by multiplying their individual probabilities of not detecting, which is \[(1 - 0.8)^3 = 0.008.\] This independence forms the basis of calculating the total detection probabilities, either for none or any other specific scenario among the radar screens.

Detection Probability

Detection probability refers to the likelihood that at least one of the radar screens detects an incoming missile.

• In the scenario, there's individual probability (0.8) for one screen to detect a missile.
• This probability influences how we calculate outcomes such as detection by at least one or more screens.

To explore different outcomes, we use the binomial distribution formula. For detecting by only one screen, the formula \[\binom{3}{1} \times (0.8)^1 \times (0.2)^{2} = 0.096\] helps us derive its probability.For detection by at least two screens, consider all scenarios where two or more screens detect the target. Calculating these involves summing probabilities from zero detection and one only, then subtracting from one:\[1 - (0.008 + 0.096) = 0.896.\] This determination ensures the radar system's reliability by understanding and improving detection mastery.